Which type of 3rd party application is this, and which SSO authentication methods does it support?

If you can find a common one, then that will be good for you.

Specifically for non-SAP systems re-using the SAP LogonTickets, I know that you can extract the user name from the ticket. I think SAP even provides some verification tools here for external applications to verify the ticket?

Currently there is much excitement about SAML 2.0 which is also worth taking a look into as well.